Maximum Maximum of Martingales given Marginals

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Maximum Maximum of Martingales given Marginals

Pierre Henry-Labordere, Jan Obloj, Peter Spoida, Nizar Touzi

To cite this version:

Pierre Henry-Labordere, Jan Obloj, Peter Spoida, Nizar Touzi. Maximum Maximum of Martingales

given Marginals. 2013. �hal-00684005v2�

Maximum Maximum of Martingales given Marginals

Pierre Henry-Labord`ere

^∗

Jan Ob l´oj

^†

Peter Spoida

^‡

Nizar Touzi

^§

April 2013

Abstract

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We present a general du- ality result which converts this problem into a min-max calculus of variations problem where the Lagrange multipliers correspond to the static part of the hedge. Following Galichon, Henry-Labord`ere and Touzi [19], we apply stochastic control methods to solve it explicitly for Lookback options with a non-decreasing payoff function. The first step of our solution recovers the extended optimal properties of the Az´ema-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek [22]

(under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers [9].

The robust superhedging cost is complemented by (simple) dynamic trading and leads to a class of semi-static trading strategies. The superhedging property then reduces to a functional inequality which we verify independently. The optimality follows from existence of a model which achieves equality which is obtained in Ob l´ oj and Spoida [33].

Key words: Optimal control, robust pricing and hedging, volatility uncertainty, optimal transportation, pathwise inequalities, lookback option.

AMS 2010 subject classifications: Primary 91G80, 91G20; secondary 49L25, 60J60.

∗Soci´et´e G´en´erale, Global Market Quantitative Research, pierre.henry-labordere@sgcib.com

†University of Oxford, Mathematical Institute, the Oxford-Man Institute of Quantitative Finance and St John’s College, Jan.Obloj@maths.ox.ac.uk

‡University of Oxford, Mathematical Institute and the Oxford-Man Institute of Quantitative Finance, Peter.Spoida@maths.ox.ac.uk

§Ecole Polytechnique Paris, Centre de Math´ematiques Appliqu´ees, nizar.touzi@polytechnique.edu

1 Introduction

The classical framework underpinning much of the quantitative finance starts by postulat- ing a probabilistic model for future prices of risky assets. The models, from their origins in Samuelson [36], Merton [28] and Black and Scholes [6] to the present day, have seen a remarkable evolution and ever increasing sophistication. Nevertheless, the essence remained the same: no arbitrage ensures that under an equivalent probability measure (discounted) asset prices are martingales and that the fair price of a future payoff is given by the cap- ital needed to replicate that payoff. That capital is then computed as the (risk-neutral) expectation of the payoff.

The classical framework has been very influential both in terms of its impact on academic research as well as on the financial industry. However, as every modelling framework, it has its important limitations. The fundamental criticism is related to the distinction between risk and uncertainty dating back to Knight [25]. The classical approach starts by postulating a stochastic universe (Ω, F , P ) which is meant to model a financial environment and capture its riskiness. What it fails to capture however is the uncertainty in the choice of P , i.e. the possibility that the model itself is wrong, also called the Knightian uncertainty. To account for model uncertainty it is natural to consider simultaneously a whole family { P

_α

: α ∈ A}

of probability measures. When all P

_α

are absolutely continuous w.r.t. one reference measure P we speak of drift uncertainty or dominated setting. This has important implications for portfolio choice problems, see F¨ollmer, Schied and Weber [18], but is not different from an incomplete market setup in terms of option pricing. However the non-dominated setup when P

_α

may be mutually singular posed new challenges and was investigated starting with Avellaneda et al. [1] and Lyons [26], through Denis and Martini [16] to several recent works e.g. Peng [34], Soner, Touzi and Zhang [37], see also [19] and the references therein.

Naturally as one relaxes the classical setup one has to abandon its precision: under model uncertainty we do not try to have a unique price but rather to obtain an interval of no- arbitrage prices. Its bounds are given by seller’s and buyer’s “safe” prices, the superreplica- tion and the subreplication prices, which can be enforced by trading strategies which work in all considered models. These bounds can be made more efficient by enlarging the set of hedging instruments. Indeed, in the financial markets certain derivatives on the underlying we try to model are liquid and have well defined market prices. Without one fixed model, these options can be included in traded asset without creating an arbitrage opportunity. By allowing to trade dynamically in the underlying and statically (today) in a range of options one hopes to have a more efficient approach with smaller intervals of possible no-arbitrage prices. This constitutes the basis of the so-called robust approach to pricing and hedging.

We contribute to this literature. Our objective here is to derive in an explicit form the

superhedging cost of a Lookback option given that the underlying asset is available for

frictionless continuous-time trading, and that European options for all strikes are available for trading for a finite set of maturities. In a zero interest rate financial market, it essentially follows from the no-arbitrage condition, as observed by Breeden and Litzenberger [8], that these trading possibilities restrict the underlying asset price process to be a martingale with given marginals. Since a martingale can be written as a time changed Brownian motion, and the maximum of the processes is not altered by a time change, the one-marginal constraint version of this problem can be converted into the framework of the Skorokhod embedding problem (SEP). This observation is the starting point of the seminal paper by Hobson [20]

who exploited the already known optimality result of the Az´ema-Yor solution to the SEP and, more importantly, provided an explicit static superhedging strategy. This methodology was subsequently used to derive robust prices and super/sub-hedging strategies for barrier options in Brown, Hobson and Rogers [10], for options on local time in Cox, Hobson and Ob l´oj [12], for double barrier options in Cox and Ob l´oj [13, 14] and for options on variance in Cox and Wang [15], see Ob l´oj [32] and Hobson [21] for more details.

The above works focused on finding explicitly robust prices and hedges for an option maturing at T given market prices of call/put options co-maturing at T . For lookback options, an extension to the case where prices at a further intermediate maturity are given can be deduced from Brown, Hobson and Rogers [9]. More recently, Hobson and Neuberger [23] treated forward starting straddle also using option prices at two maturities. Otherwise, and excluding the trivial cases when intermediate laws have no constraining effect (see e.g.

the iterated Az´ema-Yor setting in Madan and Yor [27]), we are not aware of any explicit robust pricing/hedging results when prices of call options for several maturities are given.

The most likely reason for this is that the SEP-based methodology pioneered in Hobson [20] starts with a good guess for the superhedge/embedding and these become much more difficult when more marginals are involved.

Our approach is to exploit a duality transformation which converts our problem into a martingale transportation problem: maximize the expected coupling defined by the payoff so as to transport the Dirac measure along the given distributions µ

1

, . . . , µ

n

by means of a continuous-time process restricted to be a martingale. This approach was simultaneously suggested by [4] in the discrete-time case, and [19] in continuous-time. We refer to Bonnans and Tan [7] for a numerical approximation in the context of variance options, and Tan and Touzi [39] for a general version of the optimal transportation problem under controlled dynamics.

Our general duality result converts the original problem into a min-max calculus of varia- tions problem where the Lagrange multipliers encode the intermediate marginal constraints.

An important financial interpretation is that the multiplier represent the optimal static po-

sition in Vanilla options so as to reduce the risk induced by the derivative security. Following

Galichon, Henry-Labord`ere and Touzi [19], we apply stochastic control methods to solve the

new problem explicitly. The first step of our solution recovers the extended optimal prop- erties of the Az´ema-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek [22] (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers [9]. However the stochastic control only allows us to prove an upper bound on the superreplication price. To show that the bound is optimal we need to construct a model which fits the given marginals and attains the bound. To do this we revert to the SEP methodology.

Equipped with a candidate for the static position in the optimal hedge we are able to guess the corresponding dynamic counterpart and obtain a class of semi-static trading strategies.

The superhedging property then reduces to a functional inequality which we verify indepen- dently. The optimality then follows from existence of a model which achieves equality and which is derived from SEP results obtained in Ob l´oj and Spoida [33].

The paper is organized as follows. Section 2, provides the precise mathematical formulation of the problem and states the main pricing/hedging duality result for arbitrary measurable claims under n-marginal constraints. It also discusses the link with martingale optimal transport. Our main result is given in Section 3. Pathwise arguments, including the su- perreplicating strategy, provide a first self-contained proof of the main theorem and are reported in Section 4. The stochastic control approach which allowed us to guess the correct quantities for the pathwise arguments, is reported in Section 5. Additional arguments for the one marginal case are given in Section 6. A proof of one technical lemma is relegated to the Appendix.

2 Robust superhedging of Lookback options

2.1 Modeling the volatility uncertainty

The probabilistic setting is the same as in [19] and we introduce it briefly. Further, we limit ourselves to a one-dimensional setting which is mostly relevant here.

Let Ω

x

:= {ω ∈ C([0, T ], R

¹

) : ω

0

= x} and write Ω := Ω

0

. Consider Ω as the canonical space equipped with the uniform norm kωk

_∞

:= sup

_0≤t≤T

|ω

t

|, B the canonical process, P

₀

the Wiener measure, F := {F

_t

}

_0≤t≤T

the filtration generated by B. Throughout the paper, X

0

is some given initial value in R , and we denote

X

t

:= X

₀

+ B

t

for t ∈ [0, T ].

For all F −progressively measurable processes σ with values in R

⁺

and satisfying R

T

0

σ

²_s

ds <

∞, P

₀

−a.s., we define the probability measures on (Ω, F ):

P

^σ

:= P

₀

◦ (X

^σ

)

⁻¹

where X

_t^σ

:= X

0

+ Z

t

0

σ

r

dB

r

, t ∈ [0, T ], P

₀

− a.s.

so that X is a P

^σ

−local martingale. P

S

denotes the collection of all such probability measures on (Ω, F ). The quadratic variation process hXi = hB i is universally defined and takes values in the set of all non-decreasing continuous functions with hBi

0

= 0. Moreover, for any P

^σ

∈ P

S

, hB i

t

is absolutely continuous with respect to the Lebesgue measure.

In this section, we shall consider a convenient subset P ⊂ P

S

, satisfying some technical conditions. For all P ∈ P, we think of (Ω, F

T

, F , P ) as a possible model for our financial market, where (X

t

) denotes the forward price of the underlying, i.e. we use the discounted units and the money market account is just constant.

The coordinate process stands for the price process of an underlying security and we focus on the situation when prices of liquidly traded options allow to back out the (risk-neutral) distribution of the underlying security, as observed by Breeden and Litzenberger [8]. To this end we shall focus on the probability measures in P ∈ P which satisfy the following requirement

¹

{X

t

: t ≤ T } is a P − uniformly integrable martingale. (2.1) For all P ∈ P , we denote by H

⁰

( P ) the collection of all ( P , F )−progressively measurable processes, and

H

²

loc

( P ) := n

H ∈ H

⁰

( P ) : R

T

0

|H

t

|

²

dhBi

t

< ∞, P − a.s. o .

Finally, throughout the paper, all functions are implicitly taken to be Borel measurable.

2.2 Robust super-hedging by trading the underlying

We consider the robust superhedging problem of some derivative security defined by the payoff ξ : Ω

X0

−→ R at some given maturity T > 0. We assume that ξ is F

T

−measurable.

Under the self-financing condition, for any portfolio process H, the portfolio value process Y

_t^H

:= Y

0

+

Z

t

0

H

s

· dB

s

, t ∈ [0, T ], (2.2)

is well-defined P −a.s., whenever H ∈ H

²

loc

( P ), for every P ∈ P . This stochastic integral should be rather denoted Y

_t^HP

to emphasize its dependence on P , see however Nutz [29].

Let ξ be an F

T

−measurable random variable. We introduce the subset of martingale measures:

P (ξ) := { P ∈ P : E

^P

[ξ

⁻

] < ∞}.

1We thus rule out cases when the coordinate process is a strict local martingale, which may be of interest in modelling financial bubbles, see e.g. Cox and Hobson [11], Jarrow, Protter and Shimbo [24].

The reason for restricting to this class of models is that, under the condition that E

^P

[ξ

⁺

] <

∞, the hedging cost of ξ under P is expected to be −∞ whenever E

^P

[ξ

⁻

] = ∞. As usual, in order to avoid doubling strategies, we introduce the set of admissible portfolios:

H(ξ) :=

H : H ∈ H

²_loc

and Y

^H

is a P − supermartingale for all P ∈ P (ξ) . The robust superhedging problem is defined by:

U

⁰

(ξ) := inf

Y

0

: ∃ H ∈ H(ξ), Y

₁^H

≥ ξ, P − a.s. for all P ∈ P (ξ) . (2.3) Theorem 2.1 in [19] gives a dual representation of U

⁰

(ξ) for an arbitrary payoff ξ satisfy- ing some uniform continuity assumptions. More recently, Neufeld and Nutz [30] relaxed the uniform continuity condition, allowing for a larger class of random variables including measurable ones. The following extension of [30], reported in [35], is better suited to our context:

Theorem 2.1 Assume sup

P∈P

E

^P

[ξ

⁺

] < ∞. Then U

⁰

(ξ) = sup

P∈P

E

^P

[ξ]. Moreover, exis- tence holds for the robust superhedging problem U

⁰

(ξ), whenever U

⁰

(ξ) < ∞.

2.3 Robust superhedging with additional trading of Vanillas

Let n be some positive integer, 0 = t

0

< . . . < t

n

= T be some partition of the interval [0, T ].

In addition to the continuous-time trading of the primitive securities, we assume that the investor can take static positions in European call or put options with all possible strikes and maturities t

1

< · · · < t

n

. The market price of the European call option with strike K ∈ R and maturity t

i

is denoted

c

i

(K), i = 1, . . . , n, and we denote c

₀

(K) := (X

₀

− K)

⁺

.

Consider a model P ∈ P which is calibrated to the market, i.e. E

^P

[(X

ti

− K )

⁺

] = c

i

(K ) for all 1 ≤ i ≤ n and K ∈ R . Differentiating in K, as observed by Breeden and Litzenberger [8], we see that

P (X

ti

> K ) = −c

^′_i

(K+) =: µ

i

((K, ∞))

is uniquely specified by the market prices and is independent of P . Let µ = (µ

1

, . . . , µ

n

) and P(µ) := P ∈ P : X

ti

∼ µ

i

, 1 ≤ i ≤ n

be the set of calibrated market models. As X is a P -martingale, the necessary and sufficient condition for P (µ) 6= ∅ is that the µ

i

’s are nondecreasing in convex order or, equivalently,

Z

|x|dµ

i

(x) < ∞, Z

xdµ

i

(x) = X

₀

, and c

_i−1

≤ c

i

for all 1 ≤ i ≤ n, (2.4)

where now c

i

(K) = R

∞

K

(x − K )dµ

i

(x). This is a direct extension of the Strassen Theorem [38]. The necessity follows from Jensen’s inequality. For sufficiency, an explicit model can be constructed using techniques of Skorokhod embeddings, see Ob l´oj [31]. In consequence, the t

j

−maturity European derivative defined by the payoff λ

i

(X

ti

) has an un-ambiguous market price

µ

i

(λ

i

) :=

Z

λ

i

dµ

i

= E

^P

[λ(X

ti

)], for all P ∈ P (µ).

The condition P(µ) 6= ∅ embodies the fact that the market prices observed today do not admit arbitrage. By this we mean that there exists a classical model in mathematical finance which admits no arbitrage (no free lunch with vanishing risk) and reprices the call options through risk neutral expectation. For that reason we sometimes refer to (2.4) as the no-arbitrage condition.

Remark 2.1 For the purpose of the present financial application, we could restrict the measures µ

i

to have support in R

₊

and P ∈ P to be such that X

t

≥ 0 P -a.s. Note however that this is easily achieved: it suffices to assume that X

₀

> 0 and c

n

(K) = X

₀

− K for K ≤ 0. Then µ

n

((K, ∞)) = 1, K < 0, and hence µ

n

([0, ∞)) = 1. Then for any P ∈ P (µ) we have X

t

= E

^P

[X

T

|F

t

] ≥ 0 P −a.s. for t ∈ [0, T ]. In particular, µ

i

([0, ∞)) = P (X

ti

≥ 0) = 1.

As it will be made clear in our subsequent Proposition 2.1, the function λ

i

will play the role of a Lagrange multiplier for the constraint X

t_i

∼ µ

i

, i = 1, . . . , n.

We denote t := (t

1

, . . . , t

n

), λ = (λ

1

, . . . , λ

n

), µ(λ) :=

n

X

i=1

µ

i

(λ

i

), λ(x

t

) :=

n

X

i=1

λ

i

(x

ti

), and ξ

^λ

(x, t) := ξ(x) − λ(x

t

) (2.5) for x ∈ C([0, T ]). The set of Vanilla payoffs which may be used by the hedger are naturally taken in the set

Λ

^µ_n

(ξ) := n

λ ∈ Λ

^µ_n

: sup

P∈P

E

^P

ξ

^λ

+

< ∞ o

, where Λ

^µ_n

:= n

λ : λ

i

∈ L

¹

(µ

i

), 1 ≤ i ≤ n o . (2.6) The superreplication upper bound is defined by:

U

_n^µ

(ξ) := inf n

Y

₀

: ∃ λ ∈ Λ

^µ_n

(ξ) and H ∈ H(ξ

^λ

), Y

^H,λ_T

≥ ξ, P − a.s. for all P ∈ P (ξ

^λ

) o , (2.7) where Y

^H,λ

denotes the portfolio value of a self-financing strategy with continuous trading H in the primitive securities, and static trading λ

i

in the t

i

−maturity European calls with all strikes:

Y

^H,λ_T

:= Y

_T^H

− µ(λ) + λ(X

^t

), (2.8)

indicating that the investor has the possibility of buying at time 0 any derivative security with payoff λ

i

(X

ti

) for the price µ

i

(λ

i

). U

_n^µ

(ξ) is an upper bound on the price of ξ necessary for absence of strong (model-independent) arbitrage opportunities: selling ξ at a higher price, the hedger could set up a portfolio with a negative initial cost and a non-negative payoff under any market scenario.

Similar to [19], the next result is a direct consequence of the robust superhedging dual formulation of Theorem 2.1.

Proposition 2.1 Assume that sup

P∈P

E [ξ

⁺

] < ∞, and let the family of probability measures µ

i

, i = 1, . . . , n be as in (2.4). Then:

U

_n^µ

(ξ) = inf

λ∈Λ^µn(ξ)

sup

P∈P

µ(λ) + E

^P

ξ − λ(X

^t

) .

Our objective in the subsequent sections is to use the last dual formulation in order to obtain a closed form expression for the above upper bound in the following special cases:

• Lookback option ξ := g(X

T

, X

_T^∗

), with X

_T^∗

:= max

t≤T

X

t

, under one-marginal con- straint n = 1, and some “monotonicity” condition of m 7−→ g(x, m);

• Lookback option ξ := φ(X

_T^∗

), for some nondecreasing function φ, under multiple marginal constraints.

The one-marginal result is reported in Section 6, and has been recently established by Hobson and Klimmek [22] under slightly different assumptions; therefore it must be viewed as an alternative approach to that of [22]. In contrast, the multiple-marginal result of Sections 4 and 5 is new to the literature, and generalizes the earlier contribution of Brown, Hobson and Rogers [9] in the two-marginal case. It also encompasses the trivial case where one can simply iterate the one-dimensional Az´ema-Yor [2] construction, see also Madan and Yor [27].

2.4 Optimal transportation and Skorokhod embedding problem

In this short section we discuss the connection of our problem to optimal transportation theory, on one the hand, and to the Skorokhod embedding problem, on the other hand.

First, by formally inverting the inf-sup in the dual formulation of Proposition 2.1, we see that U

_n^µ

(ξ) is related to the optimization problem:

sup

P∈P(µ)

E

^P

[ξ] (2.9)

which falls in the recently introduced class of optimal transportation problems under con-

trolled stochastic dynamics, see [4, 19, 39]. In words, the above problem consists in maximiz-

ing the expected transportation cost of the Dirac measure δ

{X0}

along the given marginals

µ

1

, . . . , µ

n

with transportation scheme constrained to a specific subclass of martingales. The cost of transportation in our context is defined by the path-dependent payoff ξ(x).

The validity of the equality between the value function in (2.9) and our problem U

_n^µ

(ξ) was established recently by Dolinsky and Soner [17] for Lipschitz payoff function ω 7−→ ξ(ω) and n = 1. The corresponding duality result in the discrete time framework was obtained in [4].

Note that if we can find a trading strategy Y

^H,λ_T

as in (2.8) which superreplicates ξ:

Y

^H,λ_T

≥ ξ P -a.s. for all P ∈ P (ξ

^λ

) and a P

^max

∈ P (µ) ∩ P (ξ

^λ

) such that E

^Pmax

[ξ] = Y

0

then trivially

Y

0

≤ sup

P∈P(µ)

E

^P

[ξ] ≤ U

_n^µ

(ξ) ≤ Y

0

and it follows that we have equalities throughout. This line of attack has been at the heart of the approach to robust pricing and hedging based on the Skorokhod embedding problem, as in [20, 10, 13, 14, 15]. It relies crucially on the ability to make a correct guess for the cheapest superhedge Y

^H,λ_T

. This becomes increasingly difficult when one considers information about prices at several maturities, n > 1. In this paper, we follow the above methodology in Section 4 to provide a first proof of our main result, Theorem 3.1. Sections 5–6 then provide a second proof based on stochastic control methods. The latter is longer and more involved than the former however it was in fact necessary in order to guess the right quantities for the former.

We now specialize the discussion to the case of a Lookback option ξ = G(X

t

, X

_T^∗

), for some payoff function G. By the Dambis-Dubins-Schwartz time change theorem, we may re-write the problem (2.9) as a multiple stopping problem (see Proposition 3.1 in [19]):

sup

(τ1,...,τn)∈T(µ)

E

^P0

G X

τ1

, . . . , X

τn

, X

_τ^∗_n

, (2.10)

where the T (µ) is the set of ordered stopping times τ

1

≤ . . . ≤ τ

n

< ∞ P

₀

-a.s. with X

τi

∼

^P₀

µ

i

for all i = 1, . . . , n and (X

t∧τn

) being a uniformly integrable martingale. Elements of T (µ)

are solutions to the iterated (multi-marginal) version of the so-called Skorokhod embedding

problem (SEP), cf. [31]. Here, the formulation (2.10) is directly searching for a solution to

the SEP which maximizes the criterion defined by the coupling G(x, m). Previous works

have focused mainly on single marginal constraint (n = 1). The case G(x, m) = φ(m) for

some non-decreasing function φ is solved by the so-called Az´ema-Yor embedding [2, 3, 20],

see also [19] which recovered this result by stochastic-control approach of Section 5. The

case G(x, m) was considered recently by Hobson and Klimmek [22], where the optimality of

the Az´ema-Yor solution of the SEP is shown to be valid under convenient conditions on the

function G. This case is also solved in Section 6 of the present paper with our approach,

leading to the same results than [22] but under slightly different conditions.

The case G(x

1

, . . . , x

n

, m) = φ(m) for some nonincreasing function φ is also trivially solved by τ

^AY

(µ

n

) in the special case when the single marginal solutions are naturally ordered:

τ

^AY

(µ

i

) ≤ τ

^AY

(µ

i+1

). This is called the increasing mean residual value property by Madan and Yor [27] who established in particular strong Markov property of the resulting time- changed process. The case of arbitrary measures which satisfy (2.4) for n = 2 was solved in Brown, Hobson and Rogers [9]. In this paper we consider n ∈ N .

3 Main result

Consider the Lookback option defined by the payoff ξ = φ(X

_T^∗

), for some nondecreasing function φ.

The key ingredient for the solution of the present n−marginals Skorokhod embedding problem turns out to be the following minimization problem:

C(m) := min

ζ1≤...≤ζn≤m n

X

i=1

c

i

(ζ

i

) m − ζ

i

− c

i

(ζ

_i+1

) m − ζ

i+1

1

{i<n}

for all m ≥ X

0

, (3.1) where we understand the value in (3.1) for ζ

k

< ζ

k+1

= · · · = ζ

n

= m as limit of the value ζ

k+1

= · · · = ζ

n

= ζ → m which is clearly either +∞ or is well defined in terms of the derivative of the call function at m.

For a fixed m, the minimum above is attained by some −∞ < ζ

₁^∗

≤ · · · ≤ ζ

_n^∗

≤ m. To see this, we first observe that, by taking ζ

1

= . . . = ζ

n

it follows that (3.1) simplifies to min

ζn≤m cn(ζn)

m−ζn

which is the slope of the tangent to c

n

intersecting the x-axis in m and is strictly smaller than 1. Then C(m) < 1. On the other hand, let (ζ

₁^k

, . . . , ζ

_n^k

) be a sequence which attains minimum. Notice that

^c_m−ζ^1(ζ1)

1

→ 1 as ζ

1

→ −∞ and the remaining terms in the sum in (3.1) are non-negative. Then, if we can extract a subsequence of (ζ

₁^k

)

k

converging to

−∞, we obtain by sending k → ∞ along such a subsequence that C(m) ≥ 1, a contradiction.

Hence ζ

₁^k

is bounded from below, implying that −∞ < inf

j

ζ

₁^j

≤ ζ

₁^k

≤ . . . ≤ ζ

_n^k

≤ m, thus reducing (3.1) to a minimisation problem of a continuous function in a compact subset of R

ⁿ

.

Theorem 3.1 Let φ be a non-decreasing function and assume that the no-arbitrage condi- tion (2.4) holds. Let ζ

₁^∗

(m), . . . , ζ

_n^∗

(m) be a solution to (3.1) for a fixed m. Then,

U

_n^µ

(ξ) ≤ U := φ(X

0

) +

n

X

i=1

Z

∞

X0

c

i

(ζ

_i^∗

(m))

m − ζ

_i^∗

(m) − c

i

(ζ

_i+1^∗

(m))

m − ζ

_i+1^∗

(m) 1

{i<n}

φ

^′

(m)dm. (3.2)

Moreover, there exist λ ∈ Λ

^µ_n

, explicitly given in (4.4), and trading strategies H = H

^stock

+ H

^fwd

∈ H(ξ

^λ

), explicitly given in (4.5)-(4.6), such that U = φ(X

0

) + µ(λ) and

U + λ(X

t

) − µ(λ) +

n

X

i=1

H

t_i−1

(X

ti

− X

t_i−1

) ≥ φ(X

_T^∗

) for all ω ∈ Ω

X₀

. (3.3) Assume further that µ

1

, . . . , µ

n

satisfy Assumption A in [33]. Then, equality holds in (3.2).

As explained before, we shall provide two alternative proofs of this result. The first one, reported in Section 4, consists in a short pathwise argument, based on a guess of the form of the optimal superhedging strategy for a simple one-touch barrier option, combined with the Skorokhod embedding results of [33].

The second proof, reported in Section 5, is more involved, and builds on the stochastic control approach of [19] which develops a systematic way of solving superhedging problems under marginals constraints. As in [19], the stochastic control tools provide the upper bound, and the optimality of the bound follows from the Skorokhod embedding results of [33]. It is our intention to demonstrate that the arguments we give in these sections prove the upper bound in Theorem 3.1 and hence emphasis will be put on rigour.

A further reason for reporting the stochastic control proof in detail is that in fact it was a starting point for both the pathwise arguments in Section 4 as well as for the construction of the embedding in [33]. More precisely, it allowed us to identify the static part λ of the optimal hedge. It was then possible to guess the dynamic part of the super-hedging strategy following the intuition of two-marginal (n = 2) case in [9] to trade only at the intermediate maturities and when the barrier is hit. The embedding construction was tailored as to provide a model in which the optimal super-hedge is in fact a perfect hedge, see Section 4 for details.

Remark 3.1 It follows from [33, Section 4] that if their Assumption A fails then the bound (3.1) is not necessarily optimal.

4 The pathwise approach

4.1 A trajectorial inequality

The following trajectorial inequality is the building block for robust superhedging of the

Lookback option in the n-marginal case.

Proposition 4.1 Let ω be a c` adl` ag path and denote ω

_t^∗

:= sup

_0≤s≤t

ω

s

. Then, for m > ω

0

and ζ

1

≤ · · · ≤ ζ

n

< m:

1 {

^ω∗_tn^≥m

} ≤

n

X

i=1

(ω

ti

− ζ

i

)

⁺

m − ζ

i

+ 1

ⁿ_ω∗

ti−1<m≤ω_ti^∗o

m − ω

ti

m − ζ

i

−

n−1

X

i=1

(ω

t_i

− ζ

i+1

)

⁺

m − ζ

i+1

+ 1 {

^m≤ω_ti^{∗,ζi+1≤ω}ti

}

ω

ti+1

− ω

ti

m − ζ

i+1

. (4.1)

Proof Denote by A

n

the right hand-side of (4.1), and let us prove the claim by induction.

First, in the case n = 1, the required inequality is the same as that of [9, Lemma 2.1]:

A

1

=

(ω

t₁

− ζ

₁

)

⁺

+ 1 {

^ω∗t0<m≤ω^∗_t1

} (m − ω

t₁

) m − ζ

1

≥ ω

t1

− ζ

₁

+ m − ω

t1

m − ζ

1

1 {

^m≤ωt^∗1

} ≥ 1 {

^m≤ωt^∗1

} . We next assume that A

n−1

≥ 1

ⁿ_ω∗

tn−1≥mo

for some n ≥ 2, and show that A

n

≥ 1 {

^ω∗_tn^≥m

} . We consider two cases.

Case 1: ω

_t^∗_n−1

≥ m. Then ω

^∗_tn

≥ m, and it follows from the induction hypothesis that 1 = 1 {

^ω∗_tn^≥m

} = 1

ⁿ_ω∗

tn−1≥mo

≤ A

_n−1

. In order to see that A

_n−1

≤ A

n

, we compute directly that, in the present case,

A

n

− A

n−1

= ω

tn

− ζ

n

m − ζ

n

1

{ω_tn≥ζn}

− 1 {

^ωtn−1≥ζn

}

≥ 0. (4.2)

Case 2: ω

_t^∗_n−1

< m. As (ω

^∗_t

) is non-decreasing, it follows that ω

^∗_ti

< m for all i ≤ n − 1.

With a direct computation we obtain:

A

n

= A

⁰_n

+ (ω

tn

− ζ

n

)

⁺

m − ζ

n

+1 {

^m≤ω∗tn

}

m − ω

tn

m − ζ

n

, where A

⁰_n

:=

n−1

X

i=1

(ω

ti

− ζ

i

)

⁺

m − ζ

i

− (ω

ti

− ζ

i+1

)

⁺

m − ζ

i+1

. Since m > ω

_t^∗_i

≥ ω

ti

for i ≤ n − 1, the functions ζ 7−→ (ω

ti

− ζ)

⁺

/(m − ζ) are non-increasing.

This implies that A

⁰_n

≥ 0 by the fact that ζ

i

≤ ζ

_i+1

for all i ≤ n. Then:

A

n

≥ (ω

tn

− ζ

n

)

⁺

+ 1 {

^m≤ω∗_tn

} (m − ω

tn

) m − ζ

n

≥ (ω

tn

− ζ

n

)

⁺

+ m − ω

tn

m − ζ

n

1 {

^m≤ωtn^∗

} (4.3)

≥ ω

tn

− ζ

n

+ m − ω

tn

m − ζ

n

1 {

^m≤ω_tn^∗

} = 1 {

^m≤ω_tn^∗

} . 2

4.2 Financial interpretation

We develop now a financial interpretation of the right hand side of (4.1) as a (pathwise)

superhedging strategy for a simple knock-in digital barrier option with payoff ξ = 1 {

^XT^∗≥m

} .

It consists of three elements: a static position in call options, a forward transaction (with the shortest available maturity) when the barrier m is hit and rebalancing thereafter at times t

i

. More precisely:

(i) Static position in calls:

λ

ζ

(X

t

) :=

n

X

i=1

(X

ti

− ζ

i

)

⁺

m − ζ

i

− (X

ti

− ζ

i+1

)

⁺

m − ζ

i+1

.

For 1 ≤ i < n, we hold a portfolio long and short calls with maturity t

i

and strikes ζ

i

and ζ

_i+1

respectively. This yields a “tent like” payoff which becomes negative only if the underlying exceeds level m. Note that by setting ζ

i

= ζ

_i+1

we may avoid trading the t

i

− maturity calls. For maturity t

n

we are only long in a call with strike ζ

n

.

(ii) Forward transaction if the barrier m is hit: 1

ⁿ_X∗

ti−1<m≤X_ti^∗om−X_ti m−ζi

At the moment when the barrier m is hit, say between maturities t

i^∗−1

and t

i^∗

, we enter into forward contracts with maturity t

i^∗

.

Note that the long call position with maturity t

i^∗

together with the forward then superhedge the knock-in digital barrier option, cf. (4.3). This resembles the robust semi-static hedge in the one-marginal case, cf. [9, Lemma 2.4]. All the “tent like” payoffs up to maturity t

i^∗−1

are non-negative.

(iii) Rebalancing of portfolio to hedge calendar spreads: − P

n−1

i=1

1 {

^m≤X_ti^∗,ζi+1≤X_ti

}

X_ti+1−X_ti m−ζi+1

After the barrier m was hit, we start trading at times t

i

in such a way that a potential negative payoff of the calendar spreads

^(Xti+1_m−ζ^−ζi+1)+

i+1

−

^(X_m−ζ^ti−ζi+1)+

i+1

, i

^∗

≤ i ≤ n, is offset, cf.

(4.2).

In the above (ii) and (iii) are instances of dynamic trading which is done in a self-financing way. Their combined payoff may be written as R

T

0

H

_s^ζ

dX

s

for a suitable choice of (simple) integrand H

^ζ

. Note that here ξ ≥ 0 so P (ξ) = P and H

^ζ

∈ H(ξ

^ζ

). Let Y

0

= µ(λ

ζ

) = P

n

i=1

_c

i(ζi)

m−ζi

−

^c_m−ζ^i(ζi+1)

i+1

1

{i<n}

, which is the initial cost entering into the static position in (i), see (2.5). Then

Y

_T^ζ

= Y

0

+ Z

T

0

H

_s^ζ

dX

s

+ λ

ζ

(X

^t

) − µ(λ

ζ

)

is an example of a semi-static trading strategy as in (2.8) and the inequality (4.1) now simply says that for any choice of ζ

1

≤ . . . ≤ ζ

n

< m, our strategy Y

^ζ

superreplicates ξ.

Our candidate superhedge for ξ is the cheapest among all Y

^ζ

. Its cost is given by (3.1) and corresponds to minimizers ζ

₁^∗

(m), . . . , ζ

_n^∗

(m) of the optimization problem (3.1). To prove that indeed U

_n^µ

(ξ) = µ(λ

ζ^∗

) it suffices, as observed in Section 2.4, to find one P ∈ P (µ) such that E

^P

[ξ] = µ(λ

ζ^∗

). This is done below in Section 4.3 where, under Assumption A in [33]

and using the results therein, we actually exhibit P such that ξ = Y

_T^ζ∗

P -a.s. Moreover, P is

independent of m.

We finally extend the above hedging strategy to the context of a general Lookback payoff φ(X

_T^∗

) for some nondecreasing φ. The superhedging property (3.3) is again obtained by integrating both sides of inequality 1 {

^XT^∗≥m

} ≤ Y

_T^ζ∗(m)

against φ

^′

. The resulting optimal hedging strategy is then characterised by:

λ

i

(x) :=

Z

∞

X0

(x − ζ

_i^∗

(m))

⁺

m − ζ

_i^∗

(m) − x − ζ

_i+1^∗

(m)

+

m − ζ

_i+1^∗

(m) 1

{i<n}

!

φ

^′

(m)dm (4.4)

H

_t^stock

(ω) := − Z

∞

X0

1

ⁿ_ω∗

⌊t⌋≥m,ω⌊t⌋≥ζ_ind(t)^∗ (m)o

m − ζ

_ind(t)^∗

(m) φ

^′

(m)dm (4.5)

H

_t^fwd

(ω) := − Z

∞

X0

1

ⁿ_ω∗

⌊t⌋<m,ω_t^∗≥mo

m − ζ

_ind(t)^∗

(m) φ

^′

(m)dm. (4.6)

with ⌊t⌋ := max {t

i

: t

i

< t, i < n}, ind(t) := min {i ≤ n : t

i

> t}.

The above integrals are well defined

²

and from (4.4), (4.5) and (4.6) it is more transparent what kind of integrability conditions one has to impose on φ in order to ensure admissible trading strategies and a finite superhedging cost. Note however that as long as ζ

_i^∗

(m) 6= m we have H

^stock

< ∞, H

^fwd

< ∞ and the stochastic integral R

(H

^stock

+ H

^fwd

)dX is a simple sum and hence is well defined pathwise.

4.3 First proof of Theorem 3.1

As argued above, to finish the proof of Theorem 3.1 it suffices to exhibit P

^max

∈ P(µ) such that 1 {

^X∗T≥m

} = Y

_T^ζ∗

P

^max

-a.s. for all m ≥ X

₀

. The case of general lookback payoff then follows since φ is non-decreasing.

Under Assumption A, Ob l´oj and Spoida [33] construct an iterated extension of the Az´ema–

Yor embedding for µ

₁

, . . . , µ

n

in a Brownian motion. They define functions η

i

and stopping times τ

i

, τ

₀

= 0, τ

i

:= inf{t ≥ τ

_i−1

: X

t

≤ η

i

(X

_t^∗

)}, 1 ≤ i ≤ n, such that X

τi

∼

^P₀

µ

i

and (X

_t∧τn

) is uniformly integrable. Further, they compute explicitly the distribution of X

_τ^∗_i

. We note that in fact max

j≤i

η

j

(m) = ζ

_i^∗

(m) for all m ≥ X

0

and all 1 ≤ i ≤ n. Consider a time change of X:

Z

t

:= X

_τ

i∧

τi−1∨^t−ti−1

ti−t

, for t

_i−1

< t ≤ t

i

, i = 1, . . . , n

2Indeed, the integrands are non-negative. Further, note that under Assumption A in [33] the functions ζi^∗(m) are continuous and in particular measurable but the latter can be assumed in all generality. Indeed, for a fixedmthe setF(m) of minimisers in (3.1) is closed and for any closedK⊂Rⁿ,{m:F(m)∩K6=∅}

is equal to {m : C(m) = C^K(m)} where C^K is given as C in (3.1) but with a further requirement that (ζ₁, . . . , ζn)∈K. BothC andCK can be obtained through countable pointwise minimisation of continuous functions and hence are measurable and so is{m:C(m) =C^K(m)}. Existence of a measurable selector for F now follows from Kuratowski and Ryll-Nardzewski measurable selection theorem, see e.g. [40, Thm. 4.1].

with Z

0

= X

0

and observe that (Z

t

) is a continuous, uniformly integrable, martingale on [0, t

n

] with Z

ti

= X

τi

∼

^P₀

µ

i

. In consequence, the distribution of Z, P

^max

:= P

₀

◦ (Z )

⁻¹

, is an element of P (µ). By going back to the proof of Proposition 4.1 and inspecting the cases where a strict inequality occurs one shows using the definition of τ

i

, see [33, Definition 2.2, Lemma 3.2], that for Z these are of measure zero.

5 The stochastic control approach

We now present the methodology which led us to conjecture (3.1) as the solution to the superhedging cost. Our objective in this section is to derive the upper bound of Theorem 3.1 from the dual formulation of Proposition 2.1. Our first observation is that, from the nondecrease of the payoff function φ, it follows from the monotone convergence theorem that:

U

_n^µ

(ξ) = inf

λ∈Λ^µn(ξ)

sup

P∈P^∗

µ(λ) + E

^P

[ξ − λ(X

t

)] , (5.1) where

P

^∗

:=

P ∈ P : E

^P

[X

_T^∗

] < ∞ . (5.2) In the present approach, we assume in addition that

φ ∈ C

¹

Lipschitz, bounded, Supp(φ

^′

) bounded from above, and

Z

∞

X0

c

i

ζ

_i^∗

(m)

m − ζ

_i^∗

(m) + c

i

ζ

_i+1^∗

(m)

m − ζ

_i+1^∗

(m) 1

_{i<n}

!

φ

^′

(m)dm < ∞. (5.3) We start with an essential ingredient, namely a general one-marginal construction which allows to move from (n − 1) to n marginals.

5.1 The one marginal problem

For an inherited maximum M

0

≥ X

0

, we introduce the process:

M

t

:= M

0

∨ X

_t^∗

for t ≥ 0.

The process (X, M ) takes values in the state space ∆ := {(x, m) ∈ R

²

: x ≤ m}. Our interest in this section is on the upper bound on the price of the one-marginal (n = 1) Lookback option defined by the payoff

ξ = g (X

T

, X

_T^∗

) for some g : R × R −→ R . (5.4)

Assumption A Function g : R × R −→ R is C

¹

in (x, m), Lipschitz in m uniformly in x, and g

xx

exists as a measure.

Assumption B The function x 7−→

^gm_m−x^(x,m)

is non-decreasing.

For a function λ : R −→ R , we denote g

^λ

:= g − λ, and we write simply Λ

^µ

=

λ ∈ L

¹

(µ) : sup

P∈P^∗

E

g

^λ

(X

T

, M

T

)

⁺

< ∞ (5.5)

for all probability measure µ ∈ M ( R ). Similar to Proposition 3.1 in [19], it follows from the Dambis-Dubins-Schwartz time change theorem that the model-free upper bound can be converted into:

U

^µ

(ξ) := inf

λ∈Λ^µ

sup

τ∈T^∗

µ(λ) + J(λ, τ ) where J(λ, τ ) := E

^P0

g

^λ

(X

τ

, X

_τ^∗

)

, (5.6) and T

^∗

is the collection of all stopping times τ such that

{X

t∧τ

, t ≥ 0} is a P

₀

−uniformly integrable martingale with E

^P0

X

_τ^∗

< ∞. (5.7) Then for every fixed multiplier λ ∈ Λ

^µ

, we are facing the infinite horizon optimal stopping problem

u

^λ

(x, m) := sup

τ∈T^∗

E

^P_x,m⁰

g

^λ

(X

τ

, M

τ

)

, (x, m) ∈ ∆, (5.8)

where E

^P_x,m⁰

denotes the conditional expectation operator E

^P0

[·|(X

₀

, M

₀

) = (x, m)].

Finally, the set Λ

^µ

of (5.5) translates in the present context to:

Λ

^µ

=

λ ∈ L

¹

(µ) : sup

τ∈T^∗

E

^P0

g

^λ

(X

τ

, M

τ

)

⁺

< ∞ . (5.9)

Remark 5.1 The condition E

^P0

X

_τ^∗

< ∞ is equivalent to E

^P0

X

τ

(ln X

τ

)

⁺

1

Xτ>0

< ∞, by Doob’s L

¹

-inequality.

The dynamic programming equation corresponding to the optimal stopping problem u

^λ

defined in (5.8) is:

min

u − g

^λ

, −u

xx

= 0 for (x, m) ∈ ∆

u

m

(m, m) = 0 for m ≥ 0. (5.10)

It is then natural to introduce a candidate solution for the dynamic programming equation defined by a free boundary {x = ψ(m)}, for some convenient function ψ:

v

^ψ

(x, m) = g

^λ

(x ∧ ψ(m), m) + (x − ψ(m))

⁺

g

_x^λ

(ψ(m), m) (5.11)

= g

^λ

(x, m) −

Z

x∨ψ(m)

ψ(m)

(x − ξ)g

^λ_xx

(ξ, m)dξ, 0 ≤ x ≤ m, (5.12)

i.e. v

^ψ

(., m) coincides with the obstacle g

^λ

before the exercise boundary ψ(m), and satisfies v

_xx^ψ

(., m) = 0 in the continuation region [ψ(m), m]. However, the candidate solution needs to satisfy more conditions. Namely v

^ψ

(., m) must be above the obstacle, concave in x on (−∞, m], and it needs to satisfy the Neumann condition in (5.10).

For this reason, our strategy of proof consists in first restricting the minimization in (5.6) to those multipliers λ in the set:

Λ ˆ

^µ

:=

λ ∈ Λ

^µ

: v

^ψ

concave in x and v

^ψ

≥ g

^λ

for some ψ ∈ Ψ

^λ

, (5.13) where the set Ψ

^λ

is defined in (5.16) below so that our candidate solution v

^ψ

satisfies the Neumann condition in (5.10). Namely, by formal differentiation of v

^ψ

, the Neumann condition reduces to the ordinary differential equation (ODE):

−ψ

^′

g

_xx^λ

(ψ, m) = γ (ψ, m) where γ(x, m) := (m − x) ∂

∂x

n g

m

(x, m) m − x

o

(5.14) exists a.e. in view of Assumption B. Similar to [19], we need for technical reasons to consider this ODE in the relaxed sense. Since g

^λ

is concave in x on (−∞, ψ(m)], the partial second derivative g

^λ_xx

is well-defined as a measure on R . We then introduce the weak formulation of the ODE (5.14):

ψ(m) < m for all m ∈ R , and −

Z

ψ(E)

g

_xx^λ

(., ψ

⁻¹

)(dξ) = Z

E

γ(ψ, .)(dm) for all E ∈ B( R ), (5.15) where ψ is chosen in its right-continuous version, and is non-decreasing by the concavity of g

^λ

and the non-negativity of γ implied by Assumption B. We introduce the collection of all relaxed solutions of (5.14):

Ψ

^λ

:=

ψ : R → R right-continuous and satisfies (5.15) . (5.16) Notice that the ODE (5.14), which motivates the relaxation (5.15), does not characterize the free boundary ψ as it is not complemented by any boundary condition.

Remark 5.2 For later use, we observe that (5.15) implies by direct integration that the function x 7−→ λ(x) − R

x

ψ(X0)

R

ψ⁻¹(y) X₀

gm(ψ(ξ),ξ)

ξ−ψ(ξ)

dξdy − R

x

ψ(X0)

g

x

(ξ, ψ

⁻¹

(ξ))dξ is affine.

Proposition 5.1 Let Assumptions A and B hold true. Then:

u

^λ

≤ v

^ψ

for any λ ∈ Λ ˆ

^µ

and ψ ∈ Ψ

^λ

.